Let $\Omega$ be closed unit disk of $\mathbb{R}^n$.
Consider $\alpha,\beta\in\mathbb{R}$ and $p\in(1,\infty)$ s.t. $\alpha-\beta=\frac{n}{p}$ and $\alpha<0$.
Then, when can we say that for $f\in H^\alpha(\Omega)$ and $g\in L^p(\Omega)$, $fg\in H^\beta(\Omega)$?
(I have shown the result when $p=2$ (and obviously $p=1$) and $\Omega=\mathbb{T}^n$ by basic calculation(using Fourier coefficients), but it does not seem to be generalizable.)